Question: Divide the following complex numbers. $ \dfrac{-7+22i}{3-2i}$
Solution: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${3+2i}$ $ \dfrac{-7+22i}{3-2i} = \dfrac{-7+22i}{3-2i} \cdot \dfrac{{3+2i}}{{3+2i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(-7+22i) \cdot (3+2i)} {(3-2i) \cdot (3+2i)} = \dfrac{(-7+22i) \cdot (3+2i)} {3^2 - (-2i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(-7+22i) \cdot (3+2i)} {(3)^2 - (-2i)^2} = $ $ \dfrac{(-7+22i) \cdot (3+2i)} {9 + 4} = $ $ \dfrac{(-7+22i) \cdot (3+2i)} {13} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({-7+22i}) \cdot ({3+2i})} {13} = $ $ \dfrac{{-7} \cdot {3} + {22} \cdot {3 i} + {-7} \cdot {2 i} + {22} \cdot {2 i^2}} {13} $ Evaluate each product of two numbers. $ \dfrac{-21 + 66i - 14i + 44 i^2} {13} $ Finally, simplify the fraction. $ \dfrac{-21 + 66i - 14i - 44} {13} = \dfrac{-65 + 52i} {13} = -5+4i $